1. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Opening Routine
Interpreting a Graph in a Coordinate Plane
Explain all that you can
about points A and B
on the graph below.

2. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Opening Routine
Interpreting a Graph in a Coordinate Plane
By reading the axes we see
that the graph shows us the relationship
between the number of years a person
has been in school and the number of
books that person has read.

3. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Opening Routine
Interpreting a Graph in a Coordinate Plane
Point A represents someone who has not
been in school very many years and,
consequently, has not read many books.
A way to interpret the graph is to note that
point B represents a person who has been
in school about three times longer than the
person represented by point A and who has
read about three times as many books.

4. Topic IV: Linear Equations in Two Variables

5. Linear Equations in Two Variables
Interpreting graphs, Plotting points,
Midpoint formula
Objectives: Find the midpoint of a segment given both endpoints. Find the endpoint of a segment given one endpoint and the midpoint of the segment.
Essential Question: How you can find the midpoint of a segment using the coordinates of both endpoints?

6. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Vocabulary
Coordinate plane: Is formed by two perpendicular
number lines that intersect at the point corresponding to the number 0 on each line.
Origin: The point of intersection of the two number lines is called the origin.
x-axis: The horizontal number line is called the x-axis.
y-axis: The vertical number line is called the x-axis.

7. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Vocabulary
Quadrants: The x- and y-axes divide
the plane into four regions, called
quadrants, which are numbered
as shown.
x-coordinate: For a point in the plane located at (a,b), the number a is the x-coordinate of the point.
y-coordinate: For a point in the plane located at (a,b), the number b is the y-coordinate of the point.

8. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Vocabulary
Segment: A measureable part of a line that consists of two points, called endpoints, and all of the points between them.
Midpoint: The point on a segment exactly halfway between the endpoints of the segment.
Distance: Is the shortest segment between two points that indicate how far apart two points are.

9. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
When you interpreting a graph, it is important to know that every point on a graph represents two pieces of information. The two pieces of information correspond to the labels on each of the two axes. In general, a point on a graph is named or interpreted (described) by first considering the horizontal axis, then the vertical axis.

10. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
This graph shows the relationship
between the temperature of a cup
of coffee and the time since the
coffee was poured. The farther to
the right a point is, the more time
has passed since the coffee was
poured. The higher a point is
vertically, the greater the
temperature. Point A represents
a cup of coffee which was poured
recently and is still hot. Point B represents a cup of coffee that was
poured awhile ago and is cooler in temperature.

11. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
This graph shows the relationship
between the cost of the internet service
and the time spent in the internet, for
two different plans CompUs and AOH.
Point A is the base price of AOH, point B
is the base price of CompUs, and point C
is where the cost of using the internet is
the same for both plans. The graph
shows that AOH’s base price is higher that
CompUs’ base price, but the cost of the service for unit of time is higher for CompUs. It indicates that CompUs is more beneficial for
users that spent little time on the internet but AOH is better for users that use it for more time.

12. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
The algebraic signs of the
x-coordinate and the y-coordinate
of any point (x, y) in each of the four
quadrants are indicated in the figure
on the right. Points on either of the
two axes are not considered to be in
any quadrant. Since a point on the
x-axis has the form (x, 0), an equation
that describes the x-axis is y = 0.

13. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Similarly, a point on the y-axis has
the form (0, y) and so an equation
of the y-axis is x = 0. When we
locate a point in the coordinate
plane corresponding to an ordered
pair of numbers and represent it
using a solid dot, we say that we
plot or graph the point.

14. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Example 1. Plot the points A (1, 2),
B (4, 3), C (3/2, 2), D (0, 4), and
E(3.5, 0). Specify the quadrant
in which each point lies.
The five points are plotted in the
coordinate plane in the figure on the
right. Point A lies in the first quadrant
(Quadrant I), B in the second quadrant
(Quadrant II), and C is in the third
quadrant (Quadrant III). Points D and E,
which lie on the y- and the x-axes,
respectively, are not in any quadrant.

15. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Sketch the set of points (x, y) in the
xy-plane that satisfy both and
0  x  2 and y = 1.
First, recall that the absolute-value
equation y = 1 implies that y = 1 or
y = 1. Thus the points that satisfy the
given conditions are the points whose
coordinates (x, y) simultaneously satisfy
the conditions: each x-coordinate is a
number in the closed interval [0, 2] and each
y-coordinate is either y = 1 or y = 1. For example, (1, 1), (½, 1), (2, 1) are a few of the points that satisfy the two conditions. Graphically, the set of all points satisfying the two conditions are points on the two parallel line segments shown in the figure on the right.

16. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula

17. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Midpoint in a number line

18. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula

19. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula

20. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Guided Practice – WE DO
Find the other endpoint of the
segment with an endpoint in
(4, 2) and the midpoint in (2,5)

21. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Independent Practice – YOU DO
Worksheet “Interpreting graphs, Plotting points and Midpoint formula”
Exercises .

22. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Homework
Complete Worksheet “Interpreting graphs, Plotting points and Midpoint formula”
Exercises .

23. Linear Equations in Two Variables
Interpreting graphs, Plotting points, Midpoint formula
Closure
Essential Question: How you can find the midpoint of a segment using the coordinates of both endpoints? .